It depends. I'm fairly bomb proof as a rider (despite being slow), and while I don't *like* riding with 50 and 60mph traffic on a 2 lane road with no shoulder, I can do it. Other riders are more nervous, and will react badly if they're passed at high speeds. Badly in this case means doing things like steering off the road or making a panic stop... it's *really* dangerous. In some cases, the road surface might be 6-12" above the grass shoulder, so going offroad even at 15mph is a great way to get hurt. (or there might be a several foot drop into a drainage ditch)

Some cyclists are even more bombproof than me, and rather enjoy the narrow two lane roads, even when speeds are high.

Knowing where you stand on this means you'll do a better job picking routes. The way to figure it out is do a fair bit of riding. If you're a more nervous sort, you'll probably do best on low traffic routes. No matter what sort you are, state tourism departments offer some help on route choice and it's well worth consulting them (your tax dollars at work... might as well take advantage of it!). There are also bike touring organizations that sell route maps, which you might find quite helpful.